The Emergence of Benford's Distribution in Directed Networks: A Study of a Multiplicative Evolution Mechanism

The Emergence of Benford's Distribution in Directed Networks: A Study of a Multiplicative Evolution Mechanism | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article The Emergence of Benford's Distribution in Directed Networks: A Study of a Multiplicative Evolution Mechanism Jun Zhou, Zijing Wang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7818774/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Network degree distributions represent a fundamental topological characteristic of complex networks, yet the study of Benford's law in network generation models remains insufficiently explored. This paper introduces a novel Multiplicative Iterative Evolution (MIE) model that drives network evolution through iterative random multiplicative updates and topological reconnections on a fixed-size network. Simulation experiments demonstrate that the in-degree distributions of networks generated by this model converge stably and exhibit high conformity to Benford's law. The MIE model provides a new, non-growth generation paradigm for network science and offers a powerful generative mechanism for explaining potential Benford phenomena in real-world networks. Unlike traditional preferential attachment models that rely on network growth, our approach demonstrates that multiplicative processes alone can produce scale-invariant degree distributions following Benford's law through evolutionary dynamics on fixed network topologies. The model's robustness across parameter ranges and its theoretical grounding in log-uniform distributions establish it as a fundamental contribution to understanding the emergence of power-law-like phenomena in complex systems. Physical sciences/Mathematics and computing Physical sciences/Physics Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7818774","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":542878841,"identity":"8d4fdcc6-6f64-4820-834b-4d30be5f1e36","order_by":0,"name":"Jun Zhou","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA2UlEQVRIiWNgGAWjYDACCQiWg/EZG4jVYkyiFiBIhKkkrEV+dvMxCcu2O+n97cnHHvxgsJHdcID52QN8WhjnHEuTkGx7ljvjzLN0wx6GNOMNB9jMDfBpYZbIMQNqOZy7AciQZmA4nLjhAA+bBD4tbFAt6QYS+d+AWv4T1sID1ZJgIJHDBtRygLAWCYm0ZAuJc4cNgX4xN+wxSDaeeZjNDK8W+RnJB29LlB2W529PfvbgR4WdbN/x5md4tYAAM0RFAhsDAyiomAmpBwLGD3Ato2AUjIJRMAqwAAAJukN1rMGs8AAAAABJRU5ErkJggg==","orcid":"","institution":"Chongqing Electric Power College","correspondingAuthor":true,"prefix":"","firstName":"Jun","middleName":"","lastName":"Zhou","suffix":""},{"id":542878842,"identity":"6ccb9083-2c26-4b8e-a7a3-82d9d3add4ae","order_by":1,"name":"Zijing Wang","email":"","orcid":"","institution":"Hong Kong University of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Zijing","middleName":"","lastName":"Wang","suffix":""}],"badges":[],"createdAt":"2025-10-09 14:38:41","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7818774/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7818774/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":95747036,"identity":"32ae4c65-1e47-427e-9746-70f642a17285","added_by":"auto","created_at":"2025-11-12 14:57:49","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":4242,"visible":true,"origin":"","legend":"","description":"","filename":"acb000d4acd14edbb2c80f9f398a24a3.json","url":"https://assets-eu.researchsquare.com/files/rs-7818774/v1/258cf6b504aa8094fdcd240c.json"},{"id":100369667,"identity":"809658e6-469d-4ad0-b614-5c46d3d23076","added_by":"auto","created_at":"2026-01-16 07:59:15","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":754079,"visible":true,"origin":"","legend":"","description":"","filename":"TheEmergenceofBenfordsDistributioninDirectedNetworksAStudyofaMultiplicativeEvolutionMechanism3.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7818774/v1_covered_12b5f60f-9d06-4907-95e9-91bf91f39b76.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"The Emergence of Benford's Distribution in Directed Networks: A Study of a Multiplicative Evolution Mechanism","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-7818774/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7818774/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Network degree distributions represent a fundamental topological characteristic of complex networks, yet the study of Benford's law in network generation models remains insufficiently explored. 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